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A version of Dwyer-Kan localization in the context of infinity-categories and simplicial categories is presented. Some results of the classical papers by Dwyer and Kan on simplicial localization are reproven and generalized. It is proven that a Quillen pair of model categories gives rise to an adjoint pair of the DK localizations. Also a result on localization of a family of infinity-categories is proven. This, in particular, is applied to localization of symmetric monoidal infinity-categories, where some (partial) results are obtained.
In this paper, we construct a refined, relative version of the etale realization functor of motivic spaces, first studied by Isaksen and Schmidt. Their functor goes from the $infty$-category of motivic spaces over a base scheme $S$ to the $infty$-cat
We define a $K$-theory for pointed right derivators and show that it agrees with Waldhausen $K$-theory in the case where the derivator arises from a good Waldhausen category. This $K$-theory is not invariant under general equivalences of derivators,
Given a double cover $pi: mathcal{G} rightarrow hat{mathcal{G}}$ of finite groupoids, we explicitly construct twisted loop transgression maps, $tau_{pi}$ and $tau_{pi}^{ref}$, thereby associating to a Jandl $n$-gerbe $hat{lambda}$ on $hat{mathcal{G}}
The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads generated
We consider a deformation of the Robert-Wagner foam evaluation formula, with an eye toward a relation to formal groups. Integrality of the deformed evaluation is established, giving rise to state spaces for planar GL(N) MOY graphs (Murakami-Ohtsuki-Y